Ado's theorem
In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras.
Statement
Ado's theorem states that every finite-dimensional Lie algebra L over a field K of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket. More precisely, the theorem states that L has a linear representation ρ over K, on a finite-dimensional vector space V, that is a faithful representation, making L isomorphic to a subalgebra of the endomorphisms of V.
History
The theorem was proved in 1935 by Igor Dmitrievich Ado of Kazan State University, a student of Nikolai Chebotaryov.
The restriction on the characteristic was later removed by Kenkichi Iwasawa (see also the below Gerhard Hochschild paper for a proof).
Implications
While for the Lie algebras associated to classical groups there is nothing new in this, the general case is a deeper result. Applied to the real Lie algebra of a Lie group G, it does not imply that G has a faithful linear representation (which is not true in general), but rather that G always has a linear representation that is a local isomorphism with a linear group.
References
- Ado, Igor D. (1935), "Note on the representation of finite continuous groups by means of linear substitutions", Izv. Fiz.-Mat. Obsch. (Kazan') 7: 1–43. (Russian language)
- Ado, Igor D. (1947), "The representation of Lie algebras by matrices" (in Russian), Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 2 (6): 159–173, ISSN 0042-1316, http://mi.mathnet.ru/eng/umn/v2/i6/p159 translation in Ado, Igor D. (1949), "The representation of Lie algebras by matrices", American Mathematical Society Translations 1949 (2): 21, ISSN 0065-9290
- Iwasawa, Kenkichi (1948), "On the representation of Lie algebras", Japanese Journal of Mathematics 19: 405–426
- Harish-Chandra (1949), "Faithful representations of Lie algebras", Annals of Mathematics, Second Series 50: 68–76, doi:10.2307/1969352, ISSN 0003-486X
- Hochschild, Gerhard (1966), "An addition to Ado's theorem", Proceedings of the American Mathematical Society 17: 531–533, doi:10.1090/s0002-9939-1966-0194482-0, https://www.ams.org/journals/proc/1966-017-02/S0002-9939-1966-0194482-0/home.html
- Nathan Jacobson, Lie Algebras, pp. 202–203
External links
- Ado’s theorem, comments and a proof of Ado's theorem in Terence Tao's blog What's new.
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